# Higher order Finite Integration

** Fundamentals:**

The Finite Integration method represents an exact discretization of the four Maxwell equations, because the space operators **C** and **S** express simply the topological relations for the calculation of closed line- and surface integrals in the discretization grid. Approximations intervene only at the coupling between the Grid-Maxwell equations, in the so-called material matrices, as well as -in time domain- at the discretization of the time axis.

express simply the topological relations for the calculation of closed line- and surface integrals in the discretization grid. Approximations intervene only at the coupling between the Grid-Maxwell equations, in the so-called material matrices, as well as -in time domain- at the discretization of the time axis.

- The space discretization, which alone is responsible for the solution's precision in static or time-harmonic (frequency domain) field calculations.
- The time discretization, which determines the quality of the time integration method in quaistatics (implicit methods) and in transient HF field problems (mostly explicit methods), and which, together with the space discretization, determines the global convergence behavior of the method.

**Höhere Ordnung in Raum und Zeit:**

The goal of the research is the development of higher-order methods for both areas, which would allow, through an improved spatial and time resolution, the modeling of larger structures with fixed desired precision, or the increasing of the precision for a given fixed space- and time-discretization. Relatively simple test structures with known analytical solutions are used for testing the approach in time-harmonic and transient field calculations. The development and implementation are based on the object-oriented approach, which allows the arbitrary combination of the methods and represents an efficient test-platform for comparing the different tested methods.

**Strategien und Beispiele:**

*Space approach:*

Currently, different techniques for generating operators with higher-order errors are being tested.

*Time Integration:*

The focus is set on the development of higher-order integration methods for the transient field calculation. The research concentrates on the extension of the classical leap-frog approach and on the comparision to the well-known Runge-Kutta methods. The basis for the explicit method is a general solution of a differential equation in the time-discrete apace. The approximation of the intervening exponential expresseion of the system matrix through a Taylor series provides the starting point for time integration approaches of any order. The use of a single time discretization grid leads to a Runge-Kutta method of arbitrary order, while the introduction of a second time-grid, interlaced with the first by half a timestep results in an extension of the leap-frog method. Approaches for both loss-free and lossy structures are examined.

Explicit time integration schemes are characterized by a conditionally stable behavior. In order to examine the stability, such methods can be successfully represented as transmission systems. The root locus method represents, also for the transient field calculations, an universal and exact analysis method for establishing the maximal stable timestep.

**Publications on this theme:**

- H. Spachmann, R. Schuhmann, T. Weiland: Convergence, Stability and Dispersion Analysis of Higher Order Leap-Frog Schemes for Maxwell's Equations. ACES 2001, Monterey.
- H. Spachmann, R. Schuhmann, T. Weiland: Higher Order Explicit Time Integration Schemes for Maxwell's Equations. Accepted for presentation at: CEM-TD, Nottingham, Sept 2001.